Trigonometry Basic 1
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1 2 7. TRIGONOMETRIC FUNCTIONS Synopsis : 1. Let θ∈R. Take an angle of measure θ radians in the standard position. Let P(x, y) be a point on the terminal side of the angle θ such that OP = r( > 0). Then i) ii) y is called sine of θ and it is denoted by sinθ. r x is called cosine of θ and it is denoted by cosθ r iii) y (x ≠ 0) is called tangent of θ and it is denoted by tanθ. x iv) v) x (y ≠ 0) is called cotangent of θ and it is denoted by cotθ. y r (x ≠ 0) is called secant of θ and it is denoted by secθ. x vi) r (y ≠ 0) is called cosecant of θ and it is denoted by cosecθ. y These six functions (ratios) are called trigonometric functions (ratios). 2. sinθ.cosecθ = 1, sin θ = 1 , cosec θ = 1 cos ecθ 3. cosθ.secθ = 1, cos θ = 1 , sec θ = 1 sin θ sec θ cos θ 4. tanθ.cotθ = 1, tan θ = 1 , cot θ = 1 5. sin θ cos θ = tan θ, cot θ cos θ = cot θ sinθ tan θ 6. sin 2 θ + cos 2 θ = 1, sin 2 θ = 1− cos 2 θ, cos 2 θ = 1 − sin 2 θ 7. 1+tan 2 θ= sec 2 θ, tan 2 θ = sec 2 θ − 1, sec 2 θ − tan 2 θ=1.
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Transcript of Trigonometry Basic 1
7. TRIGONOMETRIC FUNCTIONS
Synopsis :1. Let R. Take an angle of measure radians in the standard position. Let P(x, y) be a point on the terminal side of the angle such that OP = r( > 0). Then
2
i)
ii)
y is called sine of and it is denoted by sin.r
x is called cosine of and it is denoted by cosr
iii) y (x 0) is called tangent of and it is denoted by tan.x
iv)
v)
x (y 0) is called cotangent of and it is denoted by cot.y
r (x 0) is called secant of and it is denoted by sec.x
vi) r (y 0) is called cosecant of and it is denoted by cosec.y
These six functions (ratios) are called trigonometric functions (ratios).
2. sin.cosec = 1, sin = 1 , cosec = 1 cos ec
3. cos.sec = 1, cos = 1 , sec = 1
sin sec
cos
4. tan.cot = 1, tan = 1 , cot = 1
5. sin cos
= tan ,
cot cos = cot sin
tan 6. sin2 + cos2 = 1, sin2 = 1 cos2, cos2 = 1 sin27. 1+tan2= sec2, tan2 = sec2 1, sec2 tan2=1.8. 1 + cot2 = cosec2 , cot2 = cosec2 1, cosec2 cot2 = 1.
9. sec + tan = 1 .sec tan
10. cosec + cot = 1 .cos ec cot 11. The values of the trigonometric functions of some standard angles :
Trigonometric Functions
0/6/4/3/23/22
sin01/21/ 23 /21010
cos13 /21/ 21/20101
12. Trigonometric functions of 2n + ; nZ1) sin(2n + ) = sin, cos(2n + ) = cos2) tan(2n + ) = tan, cot(2n + ) = cot3) sec(2n + ) = sec, cosec(2n + ) = cosec13. Trigonometric functions of (), for all values of 1) sin() = sin , 2) cos() = cos ,3) tan() = tan , 4) cot() = cot ,5) sec() = sec , 6) cosec() = cosec 14. The values of trigonometric functions of any angle can be represented in terms of an angle in the first quadrantLet A = n. where nZ, 0 . Then2 2 i) sinn. 2
= sin , if n is even
= cos , if n is odd
ii) cos n. 2
= cos , if n is even= sin , if n is odd
iii) tan n.
2
= tan , if n is even
= cot , if n is odd
iv) cot n. 2
v) sec n. 2
= cot , if n is even= tan , if n is odd= sec , if n is even
= cosec , if n is odd
vi) cosec n. 2
= cosec , if n is even= sec , if n is odd
8. COMPOUND ANGLES
Synopsis :1. i) cos (A + B) = cos A cos B sin A sin Bii) cos (A B) = cos A cos B + sin A sin B
2. i) sin (A + B) = sin A cos B + cos A sin Bii) sin (A B) = sin A cos B cos A sin B3. i) tan (A + B) = tan A + tan B 1 tan A tan Bii) tan (A B) = tan A tan B 1 + tan A tan B
4. i) cot (A + B) =
ii) cot (A B) =
cot A cot B 1 cot B + cot A
cot A cot B + 1 cot B cot A5. sin (A + B) sin (A B) = sin2A sin2B= cos2B cos2A6. cos (A + B) cos (A B) = cos2A sin2B= cos2B sin2A
7. i) tan 4
+ A
= 1 + tan A
1 tan A
= cos A + sin Acos A sin A
ii) tan 4
A =
=
1 tan A
1 + tan Acos A sin Acos A + sin A
8. i) sin (A + B + C) = (sin A cos B cos C) sin A sin B
ii) cos (A + B + C) = cos A cos B cos C
(sin A sin B cos C)
iii) tan (A + B + C) = tan A + tan B + tan C tan A tan B tan C 1 tan A tan B tan B tan C tan C tan A
9. i) sin 15 + cos 15 =
3 / 2
= sin 75 + cos 75
ii) cos 15 sin 15 = 1/ 2 = sin 75 cos 75
10. i) tan 15 + cot 15 = 4 = tan 75 + cot 75ii) cot 15 tan 15 = 2 3 = tan 75 cot 75
11. i) sin 105 = cos15 = 3 + 1
Compounds Angles2 2
ii) cos 105 = sin 15 = 1 32 2
12. i) tan 105 = cot 15 = 2 3ii) cot 105 = tan 15 = 3 2
13. i) If tan = m , tan = 1 , thenm + 1tan ( + ) = 1
2m + 1
ii) If tan =
m + 1 , tan = 1 , then tan ( ) = 1.m 2m + 1
14. i) cos cos (60 + ) cos (60 ) = 0ii) cos + cos (120 + ) + cos (120 ) = 0iii) cos + cos (240 + ) + cos (240 ) = 0iv) sin sin(60 + ) + sin (60 ) = 0v) sin + sin (120 + ) sin (120 ) = 0vi) sin + sin (240 + ) sin (240 ) = 0
15. i) If A + B = 45, then (1 + tan A) (1 + tan B) = 2 ii) If A + B = 135, then (1 tan A) (1 tan B) = 2 iii)If A + B = 225, then (1 + tan A) (1 + tan B) = 2 iv) If A + B = 45, then (1 cot A) (1 cot B) = 2v) If A + B = 135, then (1 + cot A) (1 + cot B) = 2
vi) If A + B = 225, then
(1+ cot A )(1+ cot B) = 2cot A cot B
16. i) tan (45 + ) tan (45 ) = 1ii) cot (45 + ) cot (45 ) = 1
9. MULTIPLE AND SUBMULTIPLE ANGLES
Synopsis :1. i) sin 2A = 2 sin A cos A= 2 tan A 1+ tan2 Aii) cos 2A = cos2 A sin2 A= 1 2 sin 2A= 2 cos 2A 1
2= 1 tan A1+ tan2 Aiii)tan 2A = 2 tan A 1 tan 2 A2. i) sin 3A = 3 sin A 4 sin3 Aii) cos 3A = 4 cos3 A 3 cos A3iii)tan 3A =
3 tan A tan A1 3 tan2 A3. The values of Trigonometric functions of some standard angles :
18365472
sin 5 1
410 2 5
45 + 1
410 + 2 5
4
cos10 + 2 5
45 + 1
410 2 5
45 1
4
4. i) sin A . sin (60 A) sin (60 + A) =
1 sin 3A4
ii) sin A .sin (120 A) sin (120 + A) =
1 sin 3A4
5. i) cos A . cos (60 A) cos (60 + A) =
1 cos 3A4
ii) cos A . cos (120 A) cos (120+ A) = 1 cos 3A4
6. i) tan A .tan (60 A) tan (60 + A) = tan 3A
ii) tan A .tan (120 A) tan (120 + A) = tan 3A
7. i) cot A cot (60 A) cot (60 + A) = cot 3A
ii) cot A cot (120 A) cot (120 + A) = cot 3A
8. i) sin2 + sin2 (60 + ) + sin2 (60 ) = 3/2 ii) cos2 + cos2 (60 + ) + cos2 (60 ) = 3/2 iii)sin2 + sin2 (120 + ) + sin2 (120 ) = 3/2 iv) cos2 + cos2 (120 + ) + cos2 (120 ) = 3/29. i) If tan A + tan B + tan C = tan A tan B tan C, then A + B + C = n, n Z.ii) If tan A tan B + tan B tan C + tan C tan A = 1,
then A + B + C = (2n + 1) /2, n Z.
Multiple and Submultiple Angles
10. cosx cos2x cos4x cos (2nx) =
sin(2n +1x).2n+1 sin x
10. TRANSFORMATIONS
Synopsis :1. i) sin (A + B) + sin (A B) = 2 sin A cos B ii) sin (A + B) sin (A B) = 2 cos A sin B iii)cos (A + B) + cos (A B) = 2 cos A cos Biv) cos (A B) cos (A + B) = 2 sin A sin B
C + D
C D 2. i) sin C + sin D = 2 sin
cos 2 2
C + D
C D ii) sin C sin D = 2 cos
sin 2 2
C + D
C D iii)cos C + cos D = 2 cos
cos 2 2
C + D
C D iv) cos C cos D = 2 sin
sin 2 2
C + D
D C = 2sin
sin 2 2
3. i) 2 sin A cos B = sin (A + B) + sin (A B) ii) 2 cos A sin B = sin (A + B) sin (A B) iii)2 cos A cos B = cos (A + B) + cos (A B) iv) 2 sin A sin B = cos (A B) cos (A + B)4. If cos x + cos y = a, sin x + sin y = b, then
i) tan
x + y = b
ii) sin (x + y) = 2ab 2 a a2 + b2
2 2iii)cos (x + y) =
a b a2 + b2
iv) tan (x + y) = 2ab a 2 b 2
5. If cos x cos y = a, sin x sin y = b, then
i) tan
x + y = a2 b
ii) sin (x + y) = 2ab a2 + b2
Transformation
2 2iii)cos (x + y) =
b a b2 + a2
iv) tan (x + y) = 2ab a2 b2
6. If tan x + tan y = a, cot x + cot y = b, then
i) tan (x + y) = ab b a
ii) cot (x + y) =
b aab
7. If tan x tan y = a, cot x cot y = b, then
i) tan (x y) = ab a + b
ii) cot (x y) =
a + bab
8.If A + B + C = 180, then
i) sin (A + B) = sin Cii)cos (A + B) = cos C
iii)tan (A + B) = tan Civ)cot (A + B) = cot C
+
A + B Cv) sin A B = cos C
vi) cos
= sin 2 2
2 2
+
A + B Cvii) tan A B = cot C
viii) cot
= tan 2 2
2 2
If A + B + C = 180
9. tan A + tan B + tan C = tan A tan B tan C
10. cot A cot B + cot B cot C + cot C cot A = 1
11.
tan A tan B + tan B tan C + tan C tan A = 12 2 2 2 2 2
12.
cot A + cot B + cot C = cot A cot B cot C2 2 2
2 2 2
13. sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
14. cos 2A + cos 2B + cos 2C =
1 4 cos A cos B cos C
15. cos2 A + cos2 B + cos2 C = 1 2 cos A cos B cos C
16. sin2 A + sin2 B + sin2 C = 2 + 2 cos A cos B cos C
17. sin A + sin B + sin C = 4 cos A cos B cos C
Transformation
2 2 2
18. cos A + cos B + cos C = 4 sin A sin B sin C + 12 2 2
3
11. PERIODICITY AND EXTREME VALUES
Synopsis :1. A real function f : A R is said to be a periodic function if there exists a positive real number p
such that f(x + p) = f(x), x A. The least positive real number p such that f(x + p) = f(x), x
A, is called period of f.
2. If f(x) is a periodic function with period p, then
f(ax + b) is also a periodic function with period p/|a|.
3. i) sinx is a period function with period 2. ii) cosx is a period function with period 2. iii)tanx is a period function with period . iv) cotx is a period function with period . v) secx is a period function with period 2.vi) cosecx is a period function with period 2.
4. i) The period of sinax is 2 .| a |
ii) The period of cosax is 2 .| a |
iii)The period of tanax is .| a |
1
12. TRIGONOMETRIC EQUATIONS
Synopsis :1. An equation involving trigonometric functions of a variable over a set of numbers or over a set of angles is called a trigonometric equation.2. The set of values of the variable for which the equation is satisfied is the solution set of the equation.3. The method of finding the solution set is called solving of an equation.4. If the product of trigonometric functions is zero, then each factor should be equated to zero. The solutions of the resulting equations will give the solution set of the given equation.
5. If sin = k where
1 k 1, then the value of lying in the interval [/2, /2] and satisfying the
equation is called the principal solution.
6. If cos = k where
1 k 1, then the value of lying in the interval [0, ] and satisfying the
equation is called the principal solution.7. If tan = k where kR, then the value of lying in the interval (/2, /2) and satisfying the equation is called the principal solution.8. The general solution of sin = k for principal value.
1 k 1 is = n + (1)n, nZ where [/2, /2] is the
9. The general solution of cos = k for principal value.10. If sin = 0, then = n, n Z
1 k 1 is
= 2n , nZ where (/2, /2) is the
11. If cos = 0, then = (2n + 1) / 2, n Z
12. If tan = 0, then = n, n Z
13. If (i) sin2 = sin2
ii) cos2 = cos2 and
iii) tan2 = tan2 then in each case = n , n Z
14. The number of solutions of the equation acos + bsin = c is infinite if | c |
15. The equation acos + bsin = c has no solution if c2>a2 + b2.
a2 + b2
(or) c2 a2 + b2
1
13. INVERSE TRIGONOMETRIC FUNCTIONS
Synopsis :1. The function f : [/2, /2] [1, 1] defined by f(x) = sinx is a bijection. The inverse of f from[1, 1] into [/2, /2] is also a bijection. This function is called inverse sine function or arc sinefunction. It is denoted by Sin1 or Arc sin.Now Sin1x = x = sin, x [1, 1]
2. The domains and ranges of the inverse trigonometric functions are as follows.
S.NoFunctionDomainRange
1.Sin1x[1, 1][/2, /2]
2.Cos1x[1, 1][0, ]
3.Tan1xR(/2, /2)
4.Cot1xR(0, )
5.Sec1x(, 1] U [1, ) =R(1, 1)[0, /2) U (/2, )
6.Cosec1x(, 1] U [1, ) =RR(1, 1)[/2, 0) U (0, /2)
3. i) Sin1(x) = Sin1x, for x [1, 1]ii) Cos1(x) = Cos1x, for x [1, 1]iii)Tan1(x) = Tan1x, for x R4. i) Cot1(x) = Cot1x, for x Rii) Sec1(x) = Sec1x, for (, 1] U [1, )iii)Cosec1(x) = Cosec1x, for (, 1] U [1, )5. i) Sin1(x) = Cosec1(1/x), for x [1, 0) U (0, 1] ii) Cos1(x) = Sec1(1/x), for x [1, 0) U (0, 1] iii)Tan1(x) = Cot1(1/x), for x (0, ) andTan1x = + Cot1(1/x), for x (, 0)
1
6. If x [1, 1] then Cos1x + Sin1x = /2.7. If x R, then Tan1x + Cot1x = /2.8. If x (, 1] U [1, ), thenSec1x + Cosec1x = /29. If 0x1, 0y0, then
Inverse Trigonometric Functions
i) Sin1x + Sin1y = Sin1(x
1 y2 + y
1 x2 ) , for x2 + y21
ii) Sin1x + Sin1y = Sin1(x
1 y2 + y
1 x2 ) , for x2 + y2>1
10. If 0x1, 0y1, then Sin1xSin1y = Sin1(x
1 y2 y
1 x2 )
11. If 0x1, 0y1, theni) Cos1x + Cos1y = Cos1(xy
1 x2
1 y2 ) , for x2 + y21
ii) Cos1x + Cos1y = Cos1(
1 x2
1 y2 xy) , for x2 + y2>1
12. If 0x1, 0y1, then Cos1xCos1y = Cos1(xy +
1 x2
1 y2 )
x + y 13. Tan1x+Tan1y = Tan1
, for x>0, y>0, xy0, y > 0, xy > 1. 1 xy
x + y 14. Tan1x+Tan1y = Tan1
, for x0, then Tan1xTan1y = Tan1 . 1+ xy
x y 16. If x
